Obviously, the ratio of two circular gears is constant, and while trivial, the momentary ratio is shown in Figure 4 for reference. Elliptical Gears Rotate Around the Focal Point The elliptical gear is defined by a pitch curve (centrode) as a standard ellipse. A significant kinematic difference exists in circular gears rotating around an eccentric axis: to maintain a constant center distance E while rolling without slip, a pair of identical elliptical gears must rotate about one of their foci, O1 or O2, rather than their geometric centers O, as with circular gears. This requirement stems from a geometric property of the ellipse that for any point on the pitch curve, the sum of the distances to the two foci is constant and equal to the major axis length 2 * a. When the rotation centers are placed at the foci, the sum of the instantaneous radii of the two ellipses, r1 + r2, equals the axial distance E = r1(φ) + r2(ψ), φ and ψ being the rotational angles of the two ellipses. For identical elliptical gears where one revolution of the driver corresponds to one revolution of the driven gear, the center distance is E = 2 * a, where a is the major axis length.

In polar coordinates, the elliptical centrode rotating about its focus is r(θ) = p / (1 – e * cos(θ)) with p = a * (1 - e2), a is the semi-major axis, and e = c / a, the eccentricity, and c is the distance from the ellipse center to the focal point. This rotation about the focal point results in a single, smooth speed fluctuation cycle per revolution, Figure 5. With the eccentricity chosen, the ratio spread is 10.1515 / 0.0985 = 103. The dynamics in a real-world application will be challenging. The momentary ratio course is a function of a / b (major to minor half axis), as shown below for one of the horizontal axes, Figure 6. For a / b = 1, we have a circle; the ratio is then constant at I = 1.00, see the left front edge of the color plot. In dashed red, the current design is shown. As the ellipse gets slimmer and slimmer (a / b increasing), the ratio spread goes towards several orders of magnitude (note that the vertical axis is logarithmic).